A six letters word is formed using the letters of the word LOGARITHEM with or without repetition. Find the number of words that contain exactly three different letters.

Explanation

We have letters L, O, G, A, R, I, T, H, M.
Words contain exactly three different letters.
Three letters can be selected in `.^(9)C_(3)` ways.
Now we have following cases for the occurrence of these three letters.
Case I : Occurrence of letters is 4,1,1
The letter which is occurring four times can be selected in `.^(3)C_(1)` ways.
Then letters can be arranged in `(6!)/(4!)` ways.
So, number of words in this case are `.^(3)C_(1)xx(6!)/(4!)=90`
Case II : Occurrence of letters is 3,2,1
The letter which is occurring three times can be selected in `.^(3)C_(1)` ways.
The letter which is occurring two times can be selected in `.^(2)C_(1)` ways.
Then letters can be arranged in `(6!)/(3!2!)` ways.
So, number of words in this case are `.^(3)C_(1)xx .^(2)C_(1)xx(6!)/(3!2!)=360`
Case III : Occurrence of letters is 2,2,2
Since each letters is occurring twice, number of words are `(6!)/(2!2!2!)=90`
So, total number of words `=.^(9)C_(3)xx(90+360+90)`
`=84xx540`
=45360